Introduction to Kleinian Groups

Talk by Yair Minsky

August 20, 2007

3 Hyperbolic 3-space, H , may be identified with the upper half space {(z, t) | z ∈ C, t > 0} equipped with the metric dz2 + dt2 t2 3 The isometry group of hyperbolic space Isom(H ) can be identified with the group of + 3 Mobius transformations, and the group of orientation preserving isometries Isom (H ) can be identified with PSL2(C). PSL2(C) acts on the boundary of the upper half space by a b az + b · z = c d cz + d and this action can be extended in a natural way to the interior. Non-identity elements of PSL2(C) fall into one of the following three categories: 3 -elliptic isometries (having fixed points in H ) -parabolic elements (having one fixed point on Cb, the sphere at infinity) -loxodromic elements (having two fixed points at infinity and an axis fixed set-wise):

3 A Kleinian group Γ is discrete a subgroup of Isom(H ). We will usually assume that it is orientation preserving and without elliptic elements. A discrete group is a group in

1 which the identity element is isolated. Discreteness also implies that the orbit of any 3 3 point in H is discrete, i.e. for any x ∈ H there exists a ball B containing x such that gB ∩ B = ∅ ⇐⇒ gx = x. Assuming the group has no elliptic elements, this implies that there exists a B such that gB ∩ B = ∅ for any nonidentity element. In this case 3 MΓ = H /Γ is a manifold , and it inherits a complete hyperbolic metric. Conversely, given a complete hyperbolic manifold M, there exists an isometry φ from the 3 ∗ universal cover Mf → H with φ (π1(M)) a Kleinian group, so the study of hyperbolic 3-manifolds can be reduced to the study of Kleinian groups.

Surface groups- ∼ A Kleinian group Γ is a called a surface group if Γ = π1(S) for a closed surface S. In some cases we allow compact surfaces but impose a parabolicity condition on ∂S.

3 The simplest example of a surface group sitting in H is given by considering the set 2 3 {(z, t) | z ∈ R}, an isometric copy of H sitting in H . The subgroup of the isometry group 3 of H perseving this plane is PSL2(R) ⊂ PSL2(C), so we can see hyperbolic three space as a complexiﬁcation of two dimensional hyperbolic space. Other examples of surface groups are given by considering the images of topological maps ∼ S → M whose induced map π1(S) → π1(M) is injective. Then Γ0 = π1(S) sits as a 3 3 subgroup in Γ, and we get a covering map H /Γ0 → H /Γ ≈ M.

3 H

Γ0

3 H M = Γ

Manifolds foliated by surfaces and ﬁbrations give important examples of such situations.

2 As another example of how surface groups come up in 3-manifolds, we can consider manifolds M which are homeomorphic to the interior of a compact manifold Mc. In this case surface groups are associated the the ends of the manifold. Considering the ends of a manifold allows us to get a grasp on the deformation theory of hyperbolic structures on manifolds, as we can set up a correspondence between structures on the ends of the manifold and the space of hyperbolic structures on M.

Limit sets-

3 3 Let Γ be a general Kleinian group. For any x ∈ H , Γx is a discrete set in H , but by 3 compactness it must accumulate on the boundary of H ∪ Cb = H3. The compactness of the 3 closure of H is easier to see in the Poincaréball model of hyperbolic space, which is the 3 4(dx2+dy2+dz2) unit ball in R with the metric (1−r2)2 . In this model the sphere at infinity Cb can be identified with the unit sphere. ΛΓ = Γx ∩ Cb is called the limit set of Γ. Notice that the limit set does not depend on the particular orbit we look at. If λ is a limit point of 3 Γx, γix → λ, and y is any other point in H , γix and γiy are the same distance apart for 3 all i as γi is a hyperbolic isometry. Given any segment of fixed hyperbolic length in H , its Euclidean length goes to zero as it approaches the boundary, so γix and γiy converge 3 as γix approaches ∂H .

If Γ is a finite group then ΛΓ = ∅. If Γ is infinite cyclic, the limit set consists of two points p+ and p− in the case of a group generated by a loxodromic element, or a single point in the case of a group generated by a parabolic element. The limit set can also be a single ∼ point if Γ = Z × Z is generated by two parabolics with a common fixed point.

α β

The groups listed above are all called elementary groups. If Γ is not elementary, then its limit set is uncountable. The limit set Λ can be proved to be the smallest nonempty closed Γ-invariant set in Cb, and is also the closure of the set of ﬁxed points of the parabolic and

3 loxodromic elements of the group.

As Λ is Γ-invariant, Ω = Cb\Λ is Γ-invariant. Orbits do not accumulate in Ω, and in fact Γ acts properly discontinuously on Ω, i.e. if K ⊂ Ω is compact then only ﬁnitely many of the sets γK for γ ∈ Γ intersect K. This implies that Ω/Γ is Hausdoﬀ, and it will be a manifold assuming the group does not contain elliptics.

To prove that Γ acts properly discontinuously on Ω, we introduce Cλ, the convex hull of 3 Λ, which is the smallest convex set in H whose closure in the ball contains Λ. Recall that a convex set is a set X such that for x, y ∈ X the geodesic segment containing x and y also lies in X. Note that the Γ-invariance of the limit set implies the Γ invariance of CΛ. CΛ may also be characterized as the intersection of all closed half-spaces whose extension to the sphere at inﬁnity contains λ.

3 Note that as CΛ lies in H , Γ acts properly discontinuously on CΛ. We can deﬁne a Γ- 3 3 equivariant retraction r : H ∪ ∂H → CΛ ∪ ΛΓ, by sending r(x) to the “closest point” on CΛ, where closest point is understood literally for points in the interior, and is understood to mean the point of intersection of a horoball through x for a point x on the boundary. 3 To see that this is well deﬁned, note that balls in H are strictly convex, so the following picture can’t happen:

Horoball or Sphere

Geodesic must penetrate ball

Points of Tangency

Supposed Convex Hull

4 The Γ-equivariance of this retraction is easy to see, as hitting the whole picture by an element of Γ doesn’t change the convex hull of the limit set, and if y in the closest point in CΛ to x then γy is the closest point in CΛ to γx. If K ⊂ Ω is compact, then r(K) 3 is compact, and lies in the interior of H . If γK ∩ K 6= ∅ for infinitely many γ’s, then γ(r(K)) ∩ r(K) = r(γ(K)) ∩ r(K) 6= ∅ for infinitely many γ, which is a contradiction. 3 3 We now have that Γ acts properly discontinuously on H ∪ Ω, and H ∪ Ω/Γ = M, a manifold with boundary. Ω/Γ = ∂M is called the conformal boundary at infinity: as PSL2(C) acts conformally on the boundary sphere Cb,Ω/Γ comes with the structure of a Riemann surface. Note that M is not necessarily compact.

Limit sets of Surface Groups-

2 3 Let Γ be a surface group. If Γ ⊂ PSL2(R) preserves a copy of H inside H , then in non- elementary cases the limit set is a circle, and the group is a Fuchsian group. The domain 2 of discontinuity is a disjoint union of two disks, Ω+ and Ω−, and H /Γ is conformally equivalent to both Ω+/Γ and Ω−/Γ (the retraction r is in fact conformal in the case when CΛ is a totally geodesic disc). As a more interesting case, we can consider the case when the limit set is a general Jordan curve. This is the called the quasi-Fuchsian case. A Jordan curve still separates the sphere into two discs, and in fact Ω+/Γ and Ω−/Γ will still be homeomorphic to a single surface S, and M ' S × [0, 1] ' CΛ/Γ.

+ Ω Ω+

− Ω Ω− Λ Λ Fuchsian Case quasi-Fuchsian Case

One can show that a group that is bi-Lipschitz equivalent to a Fuchsian group is quasi- Fuchsian. Furthermore, Ber’s simultaneous uniformization theorem gives that there is a one-to-one correspondence between quasi-Fuchsian groups considered up to an appropri- ate equivalence and pairs of points (X,Y ) in Teichm¨ullerspace, the space of hyperbolic structures on X and Y. A central question that will be addressed in these lectures is how information about the pair (X,Y ) gives information about the quasi-Fuchsian groups and its quotient manifold.